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An Euler diagram illustrating the association fallacy. Although A is within B and is also within C, not all of B is within C.

In notation of first-order logic, this type of fallacy can be expressed as (∃x∈S : φ(x)) → (∀x ∈ S : φ(x)), meaning "if there exists any x in the set S so that a property φ is true for x, then for allx in S the property φ must be true."

Premise A is a B

Premise A is also a C

Conclusion Therefore, all Bs are Cs

The fallacy in the argument can be illustrated through the use of an Euler diagram: "A" satisfies the requirement that it is part of both sets "B" and "C", but if one represents this as an Euler diagram, it can clearly be seen that it is possible that a part of set "B" is not part of set "C", refuting the conclusion that "all Bs are Cs".

A real-world example of guilt by association is that, in response to mass shooting incidents in the U.S. and public speculation that the perpetrators had Asperger's Syndrome, many people throughout society wrongfully stigmatized and stereotyped people with the disorder as being potentially violent and having the potential to become shooters.[citation needed] Especially after the Sandy Hook Elementary School shooting, the media (after the early reporting) and autism-advocacy organizations proffered expert opinions debunking this myth and attempting to better educate the public about autism to dissuade the stigmatization. The myth of erroneously linking Asperger's Syndrome to violence also counts as an example of the logical fallacy of questionable cause and scapegoating.[citation needed]

Guilt by association can sometimes also be a type of ad hominem fallacy, if the argument attacks a person because of the similarity between the views of someone making an argument and other proponents of the argument.[1]

This form of the argument is as follows:

Source S makes claim C.

Group G, which is currently viewed negatively by the recipient, also makes claim C.

Therefore, source S is viewed by the recipient of the claim as associated to the group G and inherits how negatively viewed it is.

An example of this fallacy would be "My opponent for office just received an endorsement from the Puppy Haters Association. Is that the sort of person you would want to vote for?"

The logical inverse of "guilt by association" is honor by association, where one claims that someone or something must be reputable because of the people or organizations that are related to it or otherwise support it. For example:

Citizens of Country X won more Nobel Prizes, gold medals, and literary awards than citizens of Country Y. Therefore, a citizen of Country X is superior to a citizen of Country Y.

In many advertisements, businesses heavily use the principle of honor by association. For example, an attractive woman will say that a specific product is good. Her attractiveness gives the product good associations.

A form of the association fallacy often used by those denying a well-established scientific or historical proposition is the so-called "Galileo Gambit." The argument goes that since Galileo was ridiculed in his time but later acknowledged to be right, that since their non-mainstream views are provoking ridicule and rejection from other scientists, they will later be recognized as correct too.[2]

This argument gained considerable public attention when it was made by Rick Perry about global warming skepticism in September 2011. Perry suggested that scientists stating "here is the fact" did not necessarily imply that this was so, and that "Galileo got outvoted for a spell."[3] The argument is flawed in that being ridiculed does not necessarily correlate with being right and that many people who have been ridiculed in history were, in fact, wrong.[4] Similarly, Daniel T. Willingham has stated that while they laughed at Galileo, "they also laughed at the Three Stooges."[5]